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%I #43 Apr 02 2024 02:57:17
%S 1,5,10,21,26,25,50,85,91,13,122,35,170,125,52,341,290,455,362,273,
%T 500,305,530,425,651,425,820,75,842,13,962,1365,1220,725,52,637,1370,
%U 905,1700,221,1682,625,1850,1281,2366,1325,2210,1705,2451,651,2900,1785
%N Numerator of sum of -2nd powers of divisors of n.
%C Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
%C Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^2*(1 - x^k)). - _Ilya Gutkovskiy_, May 24 2018
%C C. Defant proves that there are no positive integers n such that sigma_{-2}(n) lies in (Pi^2/8, 5/4). See arxiv link. - _Michel Marcus_, Aug 24 2018
%H G. C. Greubel, <a href="/A017667/b017667.txt">Table of n, a(n) for n = 1..10000</a>
%H Colin Defant, <a href="https://www.nntdm.net/papers/nntdm-21/NNTDM-21-3-80-87.pdf">On the Density of Ranges of Generalized Divisor Functions</a>, Notes on Number Theory and Discrete Mathematics, Vol. 21, No. 3 (2015), pp. 80-87; <a href="https://arxiv.org/abs/1506.05432">arXiv preprint</a>, arXiv:1506.05432 [math.NT], 2015.
%F Dirichlet g.f.: zeta(s)*zeta(s+2) [for fraction A017667/A017668]. - _Franklin T. Adams-Watters_, Sep 11 2005
%F sup_{n>=1} a(n)/A017668(n) = zeta(2) (A013661). - _Amiram Eldar_, Sep 25 2022
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017668(k) = zeta(3) (A002117). - _Amiram Eldar_, Apr 02 2024
%e 1, 5/4, 10/9, 21/16, 26/25, 25/18, 50/49, 85/64, 91/81, 13/10, 122/121, 35/24, 170/169, ...
%t Table[Numerator[DivisorSigma[-2, n]], {n, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2011 *)
%t Table[Numerator[DivisorSigma[2, n]/n^2], {n, 1, 50}] (* _G. C. Greubel_, Nov 08 2018 *)
%o (PARI) a(n) = numerator(sigma(n, -2)); \\ _Michel Marcus_, Aug 24 2018
%o (PARI) vector(50, n, numerator(sigma(n, 2)/n^2)) \\ _G. C. Greubel_, Nov 08 2018
%o (Magma) [Numerator(DivisorSigma(2,n)/n^2): n in [1..50]]; // _G. C. Greubel_, Nov 08 2018
%Y Cf. A017668 (denominator), A002117, A013661, A111003 (Pi^2/8).
%K nonn,frac
%O 1,2
%A _N. J. A. Sloane_
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